Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 9 \sqrt{3}+9 i $$

Short answer

The polar form is \(18 (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\) and the exponential form is \(18 e^{i\frac{\pi}{6}}\).

Step-by-step solution

Identify the Real and Imaginary Parts

The given complex number is \(9 \sqrt{3} + 9i\). Here, the real part is \(9 \sqrt{3}\) and the imaginary part is 9.

Plot the Complex Number

In the complex plane, plot the point \((9 \sqrt{3}, 9)\). The x-coordinate corresponds to the real part \(9 \sqrt{3}\) and the y-coordinate corresponds to the imaginary part 9.

Find the Magnitude

The magnitude (or modulus) \(r\) of the complex number is found using the formula \[ r = \sqrt{(9 \sqrt{3})^2 + 9^2} \]Simplifying further:\[ r = \sqrt{81 \cdot 3 + 81} = \sqrt{324} = 18 \]

Find the Argument

The argument (or angle) \( \theta \) is computed as follows:\[ \theta = \tan^{-1}\left(\frac{9}{9\sqrt{3}}\right) \]Simplifying:\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]

Write in Polar Form

The polar form of the complex number is \[ 18 (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \]

Write in Exponential Form

The exponential form of the complex number is \[ 18 e^{i \frac{\pi}{6}} \]

Key concepts

complex plane

The complex plane is a way to visualize complex numbers.
It's like the Cartesian coordinate system but for complex numbers. On the complex plane, the horizontal axis represents the real part, and the vertical axis represents the imaginary part.
To plot a complex number, you use its real part as the x-coordinate and its imaginary part as the y-coordinate.
So, for the complex number \(9 \sqrt{3} + 9i\), you would move \(9 \sqrt{3}\) units along the x-axis and 9 units along the y-axis to place the point.

polar form

Polar form of a complex number expresses it in terms of a radius and an angle.
Instead of using the rectangular coordinates (a, b), it uses the radius (r) and the angle (\(\theta\)).
The radius or magnitude is the distance from the origin to the point, calculated as \[ r = \sqrt{a^2 + b^2} \].
The angle is the direction from the origin to the point, calculated as \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \].
Therefore, the polar form of \(9 \sqrt{3} + 9i\) is written as \[ 18 \left(\cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right)\right) \].

exponential form

The exponential form of a complex number is a more compact way of expressing its polar form.
It employs Euler's formula: \(e^{i \theta} = \cos(\theta) + i \sin(\theta)\).
This is handy because it simplifies multiplication and division of complex numbers.
Converting from polar form to exponential form simply involves placing the radius and angle into Euler's formula.
Therefore, the exponential form of \(9 \sqrt{3} + 9i\) is \[ 18 e^{i \frac{\pi}{6}} \].

magnitude

The magnitude (or modulus) of a complex number is its distance from the origin in the complex plane.
It gives an idea of how large the complex number is and is denoted by \(r\).
To calculate the magnitude of \(9 \sqrt{3} + 9i\), we use the formula: \[ r = \sqrt{(9 \sqrt{3})^2 + 9^2} \].
Simplifying, we get \( r = 18\).
This helps in representing the complex number in polar and exponential forms.

argument

The argument of a complex number is the angle that it makes with the positive real axis.
It's an important aspect, as it tells you the direction in which the complex number lies in the complex plane.
For the number \(9 \sqrt{3} + 9i\), we calculate the argument as: \[ \theta = \tan^{-1}\left(\frac{9}{9 \sqrt{3}}\right) = \frac{\pi}{6} \].
The argument tells us the angle (measured in radians) that the complex number forms with the positive real axis.
It is crucial for converting between rectangular and polar (or exponential) forms.